Inverse Trigonometric Functions

The inverse sine function (also called arcsine and denoted arcsin) is the function sin-1 with
domain [-1, 1] and range ??
?
??
?
?
2
,
2
? ?
defined by
sin-1 x = y ? sin y = x
So sin-1 x is the number in the interval ??
?
??
?
?
2
,
2
? ?
whose sine is x.
The following relations hold true:
sin (sin-1 x) = x for -1 ? x ? 1
sin-1 (sin x) = x for
2
?
? ? x ?
2
?
The inverse cosine function (also called arccosine and denoted arccos) is the function cos-1 with
domain [-1, 1] and range [0, ?] defined by
cos-1 x = y ? cos y = x
So cos-1 x is the number in the interval [0, ?] whose cosine is x.
The following relations hold true:
cos (cos-1 x) = x for -1 ? x ? 1
cos-1 (cos x) = x for 0 ? x ? ?
The inverse tangent function (also called arctangent and denoted arctan) is the function tan-1
with domain (-?, ?) and range ??
?
??
?
?
2
,
2
? ?
defined by
tan-1 x = y ? tan y = x
So tan-1 x is the number in the interval ??
?
??
?
?
2
,
2
? ?
whose tangent is x.
The following relations hold true:
tan (tan-1 x) = x for x ??
tan-1 (tan x) = x for
2
?
? ? x ?
2
?
Section 7-4
2
Example 1: Find the exact value of each expression.
tan -1
3
? 3 sin -1 1 cos -1
2
? 2 tan -1 0
Example 2: Use a calculator to find an approximate value of each expression correct to four
decimal places.
sin -1 (1.4327) cos -1 (-0.7534) tan -1 (-2.5732) sin -1 (-0.8359)
Example 3: Find the exact value of each expression.
sin ?
?
?
?
? ?
?
?
? ?
2
2
sin 1 sin ?
?
?
?
? ?
?
?
? ?
2
3
cos 1 tan -1
? ??
?
? ??
?
??
?
??
?
?
6
sin
?
Example 4: Evaluate the expression by sketching a triangle.
sin (tan -1 5) cot (sin -1 ½ ) sin (2 cos -1 0) ?
?
?
??
?
? ? ?
5
4
cos
5
4
cos sin 1 1
Example 5: Rewrite the expression as an algebraic expression